Question 19
Let $f:[0,1] \rightarrow [0,1]$ be an increasing function and let
$$
a_{n} = \int_{0}^{1} \frac{1 + (f(x))^{n}}{1 + (f(x))^{n+1}} \, \mathrm{d}x, \quad n \in \mathbb{N}^{*}.
$$
Show that the sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ is convergent and compute its limit.
$$
a_{n} = \int_{0}^{1} \frac{1 + (f(x))^{n}}{1 + (f(x))^{n+1}} \, \mathrm{d}x, \quad n \in \mathbb{N}^{*}.
$$
Show that the sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ is convergent and compute its limit.